The numerical range of elementary operators

Abstract

For n-tuplesA=(A 1,...,A n ) andB=(B 1,...,B n ) of operators on a Hilbert spaceH, letR A,B denote the operator onL(H) defined by $$R_{A,B} (X) = \sum _{i = 1}^n A_i XB_i $$ . In this paper we prove that $$co\left\{ {\sum\limits_{i = 1}^n {\alpha _i \beta _i :(\alpha _i , \ldots ,\alpha _n ) \in W(A),(\beta _i , \ldots ,\beta _n ) \in W(B)} } \right\}^ - \subset W_0 (R_{A,B} )$$ whereW is the joint spatial numerical range andW 0 is the numerical range. We will show also that this inclusion becomes an equality whenR A,B is taken to be a generalized derivation, and it is strict whenR A,B is taken to be an elementary multiplication operator induced by non scalar self-adjoints operators.